Answer:
Here's how to find the intervals in which the function f(x) is decreasing and how to express the interval(s) using interval notation:
1. Find the derivative of the function.
The derivative of the function is f'(x) = 3(x + 1)(x - 2).
2. Identify the critical points.
The critical points are the x-values where the derivative is equal to 0 or undefined. In this case, the critical points are x = -1 and x = 2.
3. Evaluate the derivative at each interval between the critical points and the points where the function is undefined.
For the interval x < -1, let's evaluate f'(x) at x = -2: f'(-2) = 15 > 0. Since f'(x) is positive on this interval, the function is increasing.
For the interval -1 < x < 2, let's evaluate f'(x) at x = 0: f'(0) = -6 < 0. Since f'(x) is negative on this interval, the function is decreasing.
For the interval x > 2, let's evaluate f'(x) at x = 3: f'(3) = 15 > 0. Since f'(x) is positive on this interval, the function is increasing.
4. Express the intervals in interval notation.
The interval where the function is decreasing is -1 < x < 2.
Answer: Therefore, the function f(x) is decreasing over the interval (-1, 2).
I hope this explanation is helpful! Let me know if you have any other questions.
P.S. The image you sent does not contain any additional information relevant to the problem, so I did not use it in my explanation.
